Adomian Decomposition Method with Green s. Function for Solving Twelfth-Order Boundary. Value Problems

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1 Applied Mthemticl Sciences, Vol. 9, 25, no. 8, HIKARI Ltd, Adomin Decomposition Method with Green s Function for Solving Twelfth-Order Boundry Vlue Prolems Wleed Al-Hyni Deprtment of Mthemtics College of Computer Science nd Mthemtics University of Mosul, Irq Copyright 24 Wleed Al-Hyni. This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited. Astrct In this pper, the Adomin Decomposition Method with Green s function (Stndrd Adomin nd Modified Technique) is pplied to solve liner nd nonliner twelfth-order oundry vlue prolems with oundry conditions defined t even-order nd odd-order derivtives s well. The numericl results otined with smll mount of computtion re compred with the exct solutions to show the efficiency of the method. The results show tht the decomposition method is of high ccurcy, more convenient nd efficient for solving high-order oundry vlue prolems. Keywords: Adomin Decomposition Method; Adomin s polynomils; Twelfthorder oundry vlue prolems; Green s function. Introduction In the eginning of the 98 s, Adomin [-4] proposed new nd fruitful method (herefter clled the Adomin Decomposition Method or ADM) for solving liner nd nonliner (lgeric, differentil, prtil differentil, integrl, etc.) equtions. It hs een shown tht this method yields rpid convergence of the solutions series to liner nd nonliner deterministic nd stochstic equtions. A clss of chrcteristic-vlue prolems of higher order (s higher s 24) is known to rise in hydrodynmic nd hydromgnetic stility [5, 6]. Furthermore,

2 354 Wleed Al-Hyni it is widely well known tht when n infinite horizontl lyer of fluid is heted from elow nd is sujected to the ction of rottion, instility sets in. When this instility is s ordinry convection the ordinry differentil eqution is sixth order; when the instility sets in s overstility, it is modelled y n eighthorder ordinry differentil eqution. Suppose, now, tht uniform mgnetic field is lso pplied cross the fluid in the sme direction s grvity. When instility sets in now s ordinry convection, it is modelled y tenth-order oundry vlue prolem; when instility sets in s overstility, it is modelled y twelfth-order oundry vlue prolem (for detils, see [5]). Agrwl's ook [7] contins theorems which detil the conditions for existence nd uniqueness of solutions of the twelfth-order oundry vlue prolems. Different numericl nd semi nlyticl methods hve een proposed y vrious uthors to solve twelfth-order oundry vlue prolems. A few of them re; Twelfth-degree spline method [8], Modified Decomposition Method with the inverse opertor (MDM) [9], Thirteen-degree spline mehod [], Non-polynomil spline technique [], Vritionl Itertion Method (VIM) [2,5], Differentil Trnsform Method (DTM) [3] nd Homotopy Perturtion Method (HPM) [4]. The min ojective of this pper is to pply the Stndrd Adomin with Green's function (SAwGF) nd Modified Technique with Green's function (MTwGF) to liner nd nonliner twelfth-order oundry vlue prolems with oundry conditions defined t even-order nd odd-order derivtives s well. 2. Anlysis of the method Let us consider the generl BVP of twelfth-order y (2) (x) + g(x, y) = f(x), x () with oundry conditions y (i) () = α i, y (i) () = β i, i =,,2,3,4,5 (2) where y = y(x), g(x, y) is liner or nonliner function of y, nd f(x) re continuous functions defined in the intervl x [, ] nd α i, β i ; (i =,,2,3,4,5) re finite rel constnts. Applying the decomposition method s in [-4], Eq. () cn e written s Ly = f(x) Ny, where L = d2 is the liner opertor nd Ny = g(x, y) is the nonliner opertor. dx2 Consequently,

3 Adomin decomposition method with Green s function 355 y(x) = h(x) + G(x, ξ)f(ξ)dξ G(x, ξ)nydξ, (3) where h(x) is the solution of Ly = with the oundry conditions (2) nd G(x, ξ) is the Green s function [5] given y G(x, ξ) = { g 2(x, ξ) if x ξ g (x, ξ) if ξ x The Adomin s technique consists of pproximting the solution of () s n infinite series y = y n, (4) n= nd decomposing the nonliner opertor N s Ny = A n, (5) n= where A n re polynomils (clled Adomin polynomils) of y, y,, y n [-4] given y d n A n = n! dλ n [N ( λi y i )] i= λ=, n =,,2,. The proofs of the convergence of the series n= y n nd n= A n re given in [3, 7-2]. Sustituting (4) nd (5) into (3) yields y n n= = h(x) + G(x, ξ)f(ξ)dξ G(x, ξ) A n dξ. (6) n= From (6), the itertes defined using the Stndrd Adomin Method re determined in the following recursive wy:

4 356 Wleed Al-Hyni y = h(x) + G(x, ξ)f(ξ)dξ y n+ = G(x, ξ)a n dξ, n =,,2,, nd the itertes defined using the Modified Technique [22] re determined in the following recursive wy: y = h(x), y = G(x, ξ)f(ξ)dξ G(x, ξ)a dξ, y n+2 = G(x, ξ)a n+ dξ, n =,,2,. Thus ll components of y cn e clculted once the A n re given. We then define n the n-term pproximnt to the solution y y φ n [y] = i= y i with lim φ n [y] = n y. 3. Applictions nd numericl results In this section, the ADM with the Green s function (Stndrd Adomin nd Modified Technique) for solving liner nd nonliner twelfth-order oundry vlue prolems is illustrted in the following exmples. The computtions ssocited with the exmples were performed using Mple 3 pckge with precision of 4 dígits. Exmple Consider the following liner BVP of twelfth-order [,, 3, 5]: y (2) (x) + xy(x) = (2 + 23x + x 3 )e x, x (7) with oundry conditions y (i) () = i(2 i), y (i) () = i 2 e, i =,,2,3,4,5. (8) The exct solution of (7), (8) is y Exct (x) = x( x)e x. Applying the decomposition method, Eq. (7) cn e written s Ly = (2 + 23x + x 3 )e x xy(x), where L = d2 is the liner opertor. Consequently, dx2

5 Adomin decomposition method with Green s function 357 y = h(x) G(x, ξ)(2 + 23ξ + ξ 3 )e ξ dξ G(x, ξ)ξy(ξ)dξ, (9) where h(x) is the solution of Ly = with the oundry conditions (8) given y h(x) = ( e) x ( e) x + ( e) x9 8 2 ( e) x8 + ( e) x7 ( e) x x5 3 x4 2 x3 + x nd G(x, ξ) is the Green s function given y G(x, ξ) = { g 2(x, ξ) if x ξ g (x, ξ) if ξ x where g (x, ξ) = ( x x 288 x x8 52 x x6 864! ) ξ + ( x 288 x x9 234 x x x x! ) ξ + ( x x 234 x9 8 + x8 864 x x6 576 x ) ξ9 + ( x 52 x 26 + x9 864 x x7 52 x x ) ξ8 + ( x 26 + x 3456 x x8 52 x x6 576 x4 296 ) ξ7 + ( x 864 x 44 + x9 576 x x7 576 x x5 864 ) ξ6 g 2 (x, ξ) = ( ξ ξ 288 ξ ξ8 52 ξ ξ6 864! ) x + ( ξ 288 ξ ξ9 234 ξ ξ ξ ξ! ) x

6 358 Wleed Al-Hyni + ( ξ ξ 234 ξ9 8 + ξ8 864 ξ ξ6 576 ξ ) x9 + ( ξ 52 ξ 26 + ξ9 864 ξ ξ7 52 ξ ξ ) x8 + ( ξ 26 + ξ 3456 ξ ξ8 52 ξ ξ6 576 ξ4 296 ) x7 + ( ξ 864 ξ 44 + ξ9 576 ξ ξ7 576 ξ ξ5 864 ) x6. Sustituting (4) in (9), the itertes defined using the Stndrd Adomin Method re determined in the following recursive wy: y = h(x) G(x, ξ)(2 + 23ξ + ξ 3 )e ξ dξ, y n+ = G(x, ξ)ξy n (ξ)dξ, n =,,2, nd the itertes defined using the Modified Technique [22] re determined in the following recursive wy: y = h(x), y = G(x, ξ)(2 + 23ξ + ξ 3 )e ξ dξ y n+2 = G(x, ξ)ξy n+ (ξ)dξ, n =,,2,. G(x, ξ)ξy (ξ)dξ, In Tle, we list the solute errors otined y SAwGF nd MTwGF. Compring them with the Thirteen-degree spline method [], Non-polynomil spline technique [], DTM [3] nd VIM [5] results. In [] the mximum solute error is 7.38E-9 with μ =, k = 22. In [] the mximum solute error is 4.72E-6. It cn e noticed tht the result otined y the present method (SAwGF) is very superior (lower error comined with less numer of itertions) to tht otined y the other mentioned methods. From Tle, it cn e deduced tht, the error decresed monotoniclly with the increment of the integer n.

7 Adomin decomposition method with Green s function 359 Tle : Comprison of solute errors for exmple SAwGF MTwGF DTM [3] VIM [5] x n = n = 2 n = n = 2 n = E E E E E E E-5.25E E-2.37E E-2.25E E E E- 7.5E-24.73E- 3.35E E-4.47E E-.62E E- 5.38E-3.5.2E-3.89E-26.2E- 2.9E E- 8.4E E-4.49E E-.65E-23.28E-.4E E E-27 4.E- 7.4E-24.39E- 3.93E E-5.32E E-2.45E-24.23E-.23E E-6 4.8E E E E- 8.25E E- 3.27E-3 Exmple 2 Consider the following liner BVP of twelfth-order [,, 3, 5]: y (2) (x) y(x) = 2(2x cos x + sin x), x () with oundry conditions y( ) = y() =, y () ( ) = y () () = 2 sin(), y (2) ( ) = y (2) () = 4 cos() 2 sin(), y (3) ( ) = y (3) () = 6 cos() 6 sin(), y (4) ( ) = y (4) () = 8 cos() + 2 sin(), y (5) ( ) = y (5) () = 2 cos() + sin(). } () The exct solution of (), () is y Exct (x) = (x 2 ) sin x. Applying the decomposition method, Eq. () cn e written s Ly = 2(2x cos x + sin x) + y(x), where L = d2 dx2 is the liner opertor. Consequently, y = h(x) 2 G(x, ξ)(2ξ cos ξ + sin ξ)dξ + G(x, ξ)y(ξ)dξ, (2) where h(x) is the solution of Ly = with the oundry conditions () given y

8 36 Wleed Al-Hyni h(x) = ( 6 95 sin ( sin 92 + ( sin cos ) x ( cos ) x7 ( cos ) x3 ( nd G(x, ξ) is the Green s function given y G(x, ξ) = { g 2(x, ξ) if x ξ g (x, ξ) if ξ x where g (x, ξ) = ( 2 x ) ξ 587 sin cos ) x sin cos ) x sin cos ) x x x x x x ( 2 x x x x x x ) ξ + ( x x x x x x x ) ξ9 + ( x x x x x x ) ξ8 + ( x x x x x x x ) ξ7 + ( 2 x x x x x x ) ξ6 + ( 2 x x x x x x x ) ξ5 + ( x x x x x x ) ξ4 + ( x x x x x x x ) ξ3 + ( x x x x x x ) ξ2 + ( 2 x x x x x x x ) ξ + 2 x 2 x + x 8 2 x 6 + x 4 x g 2 (x, ξ) = ( 2 ξ ) x ξ ξ ξ ξ ξ ( 2 ξ ξ ξ ξ ξ ξ ) x + ( ξ ξ ξ ξ ξ ξ ξ ) x9 + ( ξ ξ ξ ξ ξ ξ ) x8 + ( ξ ξ ξ ξ ξ ξ ξ ) x7 + ( 2 ξ ξ ξ ξ ξ ξ ) x6

9 Adomin decomposition method with Green s function 36 + ( 2 ξ ξ ξ ξ ξ ξ ξ ) x5 + ( ξ ξ ξ ξ ξ ξ ) x4 + ( ξ ξ ξ ξ ξ ξ ξ ) x3 + ( ξ ξ ξ ξ ξ ξ ) x2 + ( 2 ξ 2 ξ + ξ 9 ξ ξ 5 ξ ξ ) x ξ 2 ξ + ξ 8 2 ξ 6 + ξ 4 ξ Sustituting (4) in (2), the itertes defined using the Stndrd Adomin Method re determined in the following recursive wy: y = h(x) 2 G(x, ξ)(2ξ cos ξ + sin ξ)dξ y n+ = G(x, ξ)y n (ξ)dξ, n =,,2, nd the itertes defined using the Modified Technique [22] re determined in the following recursive wy: y = h(x), y = 2 G(x, ξ)(2ξ cos ξ + sin ξ)dξ y n+2 = G(x, ξ)y n+ (ξ)dξ, n =,,2,., + G(x, ξ)y (ξ)dξ, In Tle 2, we present the solute errors otined y SAwGF nd MTwGF. Compring them with the Thirteen-degree spline method [], Non-polynomil spline technique [], DTM [3] nd VIM [5] results. In [] the mximum solute error is 4.69E-5 with μ =, k = 22. In [] the mximum solute error is 4.67E-7. It cn e noticed tht the result otined y the present method (SAwGF) is very superior to tht otined y the four previous mentioned methods. From Tle 2, it cn e deduced tht, the error decresed monotoniclly with the increment of the integer n. Tle 2: Comprison of solute errors for exmple 2 SAwGF MTwGF DTM [3] VIM [5] x n = n = 2 n = n = 2 n = E-7..22E- 8.8E E-9.52E-9.64E-5.39E E-.34E E-9 2.5E-9 2.8E E-6

10 362 Wleed Al-Hyni.3 2.2E-.45E-2 4.3E-9 2.7E E-2.65E-4.4.8E-.8E-2 3.4E E E- 2.6E-5.5.4E- 7.39E E-9.39E-9.E-.38E E E E- 6.33E E- 2.22E E E E-.86E E-.6E E-3.37E E- 2.58E E-9.26E E E-25.2E-2 6.4E E-9.99E E E-5 In results not presented here, we hve seen tht the solute errors otined y SAwGF nd MTwGF in the intervl [,] re the sme s for the intervl [,]. Exmple 3 Finlly, we consider the following nonliner BVP of twelfth-order [9, 2-4]: y (2) (x) = 2e x y 2 (x) + y (x), < x < (3) with two sets of oundry conditions y (i) () = ( ) i, y (i) () = ( ) i e, i =,,2,3,4,5 (4) y (2i) () =, y (2i) () = e, i =,,2,3,4,5. (5) The exct solution of (3) with ((4) or (5)) is y Exct (x) = e x. Applying the decomposition method, Eq. (3) cn e written s Ly = 2e x Ny + y (x), where L = d2 dx 2 is the liner opertor nd Ny = y2 is the nonliner opertor. Consequently, y = h(x) + G(x, ξ){2e ξ Ny(ξ) + y (ξ)}dξ, (6) where h(x) is the solution of Ly = with the oundry conditions (4) given y

11 Adomin decomposition method with Green s function e h(x) = x e x e + x e x e + x e x ! x5 + 4! x4 3! x3 + 2! x2 x +, nd G(x, ξ) is the Green s function given previously in exmple. For the oundry conditions (5), h(x) is given y h(x) = e x +!! x + ( e ) x9 + 8! x8 + ( 37e ) x7 + 6! x6 + ( 2863e ) x5 + 4! x4 + ( 2723e ) x3 + 2! x2 + ( e ) x +, nd G(x, ξ) is the Green s function given y G(x, ξ) = { g 2(x, ξ) if x ξ g (x, ξ) if ξ x where g (x, ξ) = (! x! ) ξ + ( x x2 + x) ξ ( 648 x5 296 x x3 x) ξ ( 648 x7 864 x x5 324 x3 + x) ξ ( x x x7 324 x5 + 7 x3 x) ξ3 2835

12 364 Wleed Al-Hyni + (! x! x x) ξ 93555! ξ 8864 x x x x3 g 2 (x, ξ) = (! ) x + ( ξ ξ2 + + ( 648 ξ5 296 ξ ξ3 ξ) x ( 648 ξ7 864 ξ ξ5 324 ξ3 + ξ) x ( ξ ξ ξ7 324 ξ5 + 7 ξ3 ξ) x (! ξ! ξ ξ) x ξ9 ξ) x ξ ξ ξ3 Sustituting (4) nd (5) in (6), the itertes defined using the Stndrd Adomin Method re determined in the following recursive wy: y = h(x), y n+ = G(x, ξ){2e ξ A n + y (ξ)}dξ, n =,,2,. For the nonliner term Ny = y 2 = n= A n the corresponding Adomin polynomils re: A = y 2, A = 2y y, A 2 = 2y y 2 + y 2, n A n = y i y n i, n i, n =,, 2,. i= In Tles 3A nd 3B we give the solute errors for the prolem (3) with oundry conditions (4) nd (5) respectively otined y SAwGF. Compring it with the MDM [9], DTM [3] nd HPM [4] results, it cn e seen esily tht the result otined y the present method (SAwGF) is very superior to tht otined

13 Adomin decomposition method with Green s function 365 y the three previous mentioned methods. From Tles 3A nd 3B, it cn e deduced tht, the error decresed monotoniclly with the increment of the integer n. Tle 3A: Comprison of solute errors for exmple 3 x SAwGF, n = DTM [3] E-6 4.E E-4.3E-3.3.E E E-3.53E E-3.98E E-3.57E-2.7.8E-3 7.7E E-4.42E E-6 4.6E E-5 Tle 3B: Comprison of solute errors for exmple 3 SAwGF MDM [9] VIM [2] DTM [3] HPM [4] x n = n = 2 n = 3 n = E-7.4E-2.6E-7.6E-7.6E E E-7 2.7E-2 3.7E-7 3.7E-7 3.7E-7.42E E E E E E-7.96E E E E E E-7 2.3E E-7 3.7E E E E E E E E E E-7 2.3E E-7 3.E E E E-7.96E E-7 2.9E-2 3.7E-7 3.7E-7 3.7E-7.42E E-7.5E-2.6E-7.6E-7.6E-7 7.5E E- 2.E-.E It is cler from the Tles 3A nd 3B tht the numericl results corresponding to prolem (3) with oundry conditions (4) re superior to those with oundry conditions (5). 4. Conclusions The ADM with Green s function (Stndrd Adomin nd Modified Technique) hs een pplied for solving liner nd nonliner twelfth-order

14 366 Wleed Al-Hyni oundry vlue prolems with oundry conditions defined t even-order nd odd-order derivtives s well. Comprison of the results otined y the present method with those otined y the Twelfth-degree spline method, Modified decomposition method with the inverse opertor, Thirteen-degree spline method, Non-polynomil spline technique, Vritionl itertion method, Differentil trnsform method nd Homotopy perturtion method hs reveled tht the present method is superior ecuse of the lower error nd fewer required itertions. It hs een shown tht error is monotoniclly reduced with the increment of the integer n. References []. G. Adomin, Stochstic Systems. Acdemic Press, New York, 983. [2]. G. Adomin, Nonliner Stochstic Opertor Equtions. Acdemic Press, New York, [3]. G. Adomin, Nonliner Stochstic Systems Theory nd Applictions to Physics. Kluwer Acdemic Pulishers, Dordrecht, [4]. G. Adomin, Solving Frontier Prolems of Physics: The Decomposition Method. Kluwer Acdemic Pulishers, Dordrecht, 994. [5]. S. Chndrsekhr, Hydrodynmic Hydromgnetic Stility. Clrendon Press, Oxford, 96 (Reprinted: Dover Books, New York, 98). [6]. K. Djidjeli, E. H. Twizell nd A. Boutye, Numericl methods for specil nonliner oundry vlue prolems of order 2m. J. Comput. Appl. Mth. 47 () (993), [7]. R. P. Agrwl, Boundry vlue prolems for higher-order differentil equtions. World Scientific, Singpore, [8]. S. S. Siddiqi nd E. H. Twizell, Spline solutions of liner twelfth-order oundry-vlue prolems. J. Comput. Appl. Mth. 78 (997), [9]. A. M. Wzwz, Approximte solutions to oundry vlue prolems of higher order y the modified decomposition method. Comput. Mth. Appl. 4 (2),

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16 368 Wleed Al-Hyni [2]. S. Guelll nd Y. Cherruult, Prcticl formul for clcultion of Adomin's polynomils nd ppliction to the convergence of the decomposition method. Int. J. Biomed. Comput. 36 (3) (994), [22]. A. M. Wzwz, A relile modifiction of Adomin decomposition method. Appl. Mth. Comput. 2 () (999), Received: Octoer 7, 24; Pulished: Jnury 5, 25